Functional Renormalization Group Study of Superconductivity in Rhombohedral Trilayer Graphene

Wei Qin, Chunli Huang, Tobias Wolf, Nemin Wei, Igor Blinov, and Allan H. MacDonald

Phys. Rev. Lett. 130, 146001 – Published 4 April 2023

Abstract

We employ a functional renormalization group approach to ascertain the pairing mechanism and symmetry of the superconducting phase observed in rhombohedral trilayer graphene. Superconductivity in this system occurs in a regime of carrier density and displacement field with a weakly distorted annular Fermi sea. We find that repulsive Coulomb interactions can induce electron pairing on the Fermi surface by taking advantage of momentum-space structure associated with the finite width of the Fermi sea annulus. The degeneracy between spin-singlet and spin-triplet pairing is lifted by valley-exchange interactions that strengthen under the RG flow and develop nontrivial momentum-space structure. We find that the leading pairing instability is $d$ -wave-like and spin singlet, and that the theoretical phase diagram versus carrier density and displacement field agrees qualitatively with experiment.

Received 5 April 2022
Revised 1 October 2022
Accepted 16 March 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.146001

Physics Subject Headings (PhySH)

Pairing mechanisms Spin-singlet pairing Spin-triplet pairing Superconductivity

Graphene

BCS theory Landau-Ginzburg theory Renormalization group

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Wei Qin ^1,*, Chunli Huang ^1,2, Tobias Wolf¹, Nemin Wei¹, Igor Blinov¹, and Allan H. MacDonald^1,†

¹Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA
²Theoretical Division, T-4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

^*weiqin@utexas.edu
^†macd@physics.utexas.edu

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Issue

Vol. 130, Iss. 14 — 7 April 2023

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Images

Figure 1
(a) Patching scheme for the annular Fermi surfaces (solid curves). The momentum space around each valley is divided into 48 patches. The circles on the Fermi surfaces specify the wave vectors at which we evaluate 4PVs. The dotted curves identify the wave vectors at which energy maxima occur along radial directions, and separate patches that belong to the inner and outer Fermi surfaces in our FRG calculations. (b) Diagrammatic representation of the one-loop RG flow equation for 4PVs, including particle-particle (PP), exchange (EX), and forward scattering (FS) contributions. Here the temperature $T$ is the RG flow time, and the slashes on the propagators denote temperature derivatives [44].
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Figure 2
(a) Schematic diagrams of $u_{a, t, e}$ . (b)–(c) Renormalized pairing interactions at temperature $T = 70 mK$ , including (b) $u_{t} (n_{1}, {\bar{n}}_{1}; {\bar{n}}_{2})$ and (c) $u_{e} (n_{1}, {\bar{n}}_{1}; n_{2})$ . The bottom-left and top-right $24 \times 24$ blocks are scatterings on the inner and outer Fermi surfaces, respectively. The remaining blocks are scatterings between the inner and outer Fermi surfaces.
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Figure 3
(a) Temperature flows of pairing interaction in channels distinguished by gap function symmetries. (b) Magnitudes of spin-singlet $d \pm i d$ and spin-triplet $p \pm i p$ gap functions on the Fermi surfaces. (c) Angular momentum amplitudes $Δ_{m}$ of the gap function on the inner (left) and outer (right) Fermi surfaces. (b)–(c) are calculated at $T = 70 mK$ . (d) Spin-singlet $d - i d$ and spin-triplet $p - i p$ gap functions on the $K$ -valley Fermi surfaces, where the length and direction of arrows specify gap amplitude and phase. (e) Schematic diagram of the inner-, inter-, and outer-Fermi surface scatterings that constitute the total pairing interaction $u_{t} (n_{1}, {\bar{n}}_{1}; {\bar{n}}_{2}, n_{2})$ . Similar definitions are used for $u_{e} (n_{1}, {\bar{n}}_{1}; n_{2}, {\bar{n}}_{2})$ . (f)–(g) Contributions from the three types of scatterings illustrated in (e) to pairing interactions in (f) spin-singlet $d$ -wave and (g) spin-triplet $p$ -wave channels.
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Figure 4
(a)–(b) EX diagram contributions to $u_{t} (n_{1}, {\bar{n}}_{1}; {\bar{n}}_{2}, n_{2})$ and $u_{e} (n_{1}, {\bar{n}}_{1}; n_{2}, {\bar{n}}_{2})$ , where blue (red) solid lines denote $K (K^{'})$ -valley propagators. (c)–(d) Particle-hole susceptibilities associated with (a)–(b), calculated at $T = 70 mK$ . (e) Three equivalent inter-Fermi-surface nesting wave vectors $k_{25} - k_{1} \approx k_{4} - k_{9} \approx k_{22} - k_{17}$ , at which $Π_{K K}^{p h}$ peaked: $(n_{1}, n_{2}) = (1, 25)$ , (9,4), and (17,22) marked by stars in (d). (f) Circular annular Fermi surfaces: $k_{F 1, 2}$ are the two Fermi wave vectors. $θ_{1, 2}$ are the angles spanned by $k_{F 2} - k_{F 1}$ on the inner and outer Fermi surfaces.
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Figure 5
Phase diagrams as function of (a) $n_{e}$ and (b) $Δ_{d}$ . The inserts depict the annular Fermi surfaces for several typical values of $n_{e}$ and $Δ_{d}$ that are associated with the SC, IVS and FM states. These results are obtained by choosing dielectric constant $ε = 4$ and gate-sample distance $d_{s} = 40 nm$ .
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